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PrimeOmega
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PrimeOmega

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gives the number of prime factors counting multiplicities in n.

Details and Options

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • PrimeOmega gives the number of prime factors of an integer with multiplicity.
  • For a number with a unit and primes, PrimeOmega[n] returns k1+⋯+km.
  • With the setting GaussianIntegers->True, PrimeOmega gives the number of Gaussian prime factors.
  • PrimeOmega[m+In] automatically works over Gaussian integers.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Compute PrimeOmega at 30:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-bnjg83
Out[1]=1

Plot the PrimeOmega sequence for the first 100 numbers:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-hiieto
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Numerical Evaluation  (4)

PrimeOmega works over integers:

In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-h7ou6e
Out[2]=2

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-k04y6h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-f359qp
Out[2]=2

Compute for large integers:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-bdyuaq
Out[1]=1

PrimeOmega threads over lists:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-ccdboa
Out[1]=1

Symbolic Manipulation  (4)

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-bkedy5

Reduce expressions:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-5or6fl
Out[1]=1

Solve equations:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-6t6jbk
Out[1]=1

Identify the PrimeOmega sequence:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-n7lgc3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-nc8uno
Out[2]=2

Options  (1)Common values & functionality for each option

GaussianIntegers  (1)

Compute PrimeOmega over integers:

In[3]:=3
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https://wolfram.com/xid/0i1m3iaz6-10e1vr
Out[3]=3

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-s7a4h8
Out[1]=1

Applications  (6)Sample problems that can be solved with this function

Basic Applications  (2)

Table of the values of PrimeOmega for the integers up to 100:

In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-es2o9g
Out[2]=2

Histogram of the values of PrimeOmega:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-eb7ks0
Out[1]=1

Number Theory  (4)

Use PrimeOmega to test for a prime number:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-ok5wcm
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-c8svys
Out[2]=2
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-do9kut
Out[2]=2

Use PrimeOmega to compute LiouvilleLambda:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-c8c4az
Out[1]=1

Compare with:

In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-ljtv55
Out[2]=2

Plot the average over values of PrimeOmega for different ranges of integer arguments:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-q1qvtp
Out[1]=1

The Fourier statistics of the PrimeOmega sequence:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-j45sop
Out[1]=1

Properties & Relations  (5)Properties of the function, and connections to other functions

Use FactorInteger to find the number of prime factors counting multiplicities:

In[4]:=4
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https://wolfram.com/xid/0i1m3iaz6-n4k1tb
Out[4]=4
In[8]:=8
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https://wolfram.com/xid/0i1m3iaz6-j3ychv
Out[8]=8

PrimeOmega is a completely additive function:

In[1]:=1
✖
https://wolfram.com/xid/0i1m3iaz6-kpqkx
Out[1]=1

PrimeOmega gives the exponent for a prime power:

In[1]:=1
✖
https://wolfram.com/xid/0i1m3iaz6-bdxu78
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-d0shx
Out[2]=2

PrimeOmega and PrimeNu are equivalent when the argument is square-free:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-wahbq
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-dvmtii
Out[2]=2

PrimeOmega is always greater than or equal to PrimeNu:

In[1]:=1
✖
https://wolfram.com/xid/0i1m3iaz6-e9gw6
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

PrimeOmega is not defined at 0:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-eh5cu1
Out[1]=1

Neat Examples  (2)Surprising or curious use cases

Plot the arguments of the Fourier transform of PrimeOmega:

In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-jeg3fb
Out[2]=2

Plot the Ulam Spiral of PrimeOmega:

In[1]:=1
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https://wolfram.com/xid/0i1m3iaz6-zp3c6
In[2]:=2
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https://wolfram.com/xid/0i1m3iaz6-qhm0kv
Out[2]=2
Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.
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Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

Text

Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

✖
Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

CMS

Wolfram Language. 2008. "PrimeOmega." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeOmega.html.

✖
Wolfram Language. 2008. "PrimeOmega." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeOmega.html.

APA

Wolfram Language. (2008). PrimeOmega. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeOmega.html

✖
Wolfram Language. (2008). PrimeOmega. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeOmega.html

BibTeX

@misc{reference.wolfram_2025_primeomega, author="Wolfram Research", title="{PrimeOmega}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeOmega.html}", note=[Accessed: 01-May-2025 ]}

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@misc{reference.wolfram_2025_primeomega, author="Wolfram Research", title="{PrimeOmega}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeOmega.html}", note=[Accessed: 01-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_primeomega, organization={Wolfram Research}, title={PrimeOmega}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeOmega.html}, note=[Accessed: 01-May-2025 ]}

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@online{reference.wolfram_2025_primeomega, organization={Wolfram Research}, title={PrimeOmega}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeOmega.html}, note=[Accessed: 01-May-2025 ]}